Workshop on GNSS Data Application to Low Latitude Ionospheric Research May Fundamentals of Satellite Navigation

Transcription

1 Workshop on GNSS Data Application to Low Latitude Ionospheric Research 6-17 May 2013 Fundamentals of Satellite Navigation HEGARTY Christopher The MITRE Corporation 202 Burlington Rd. / Rte 62 Bedford MA U.S.A.

2 Fundamentals of Satellite Navigation Chris Hegarty May

3 Chris Hegarty The MITRE Corporation (Tel) The contents of this material reflect the views of the author. Neither the Federal Aviation Administration nor the Department of the Transportation makes any warranty or guarantee, or promise, expressed or implied, concerning the content or accuracy of the views expressed herein. 2

4 Fundamentals of Satellite Navigation Geodesy Time and clocks Satellite orbits Positioning 3

5 Earth Centered Inertial (ECI) Coordinate System Oblateness of the Earth causes direction of axes to move over time So that coordinate system is truly inertial (fixed with respect to stars), it is necessary to fix coordinates J2000 system fixes coordinates at 11:58: hours UTC on January 1,

6 Precession and Nutation Vega Polaris Earth 1.Rotation axis (now pointing near Polaris) 2. In ~13,000 yrs, rotation axis will point near Vega 3. In ~26,000 yrs, back to Polaris Precession is the large (23.5 deg half-angle) periodic motion Nutation is a superimposed oscillation (~9 arcsec max) 5

7 Earth Centered Earth Fixed (ECEF) Coordinate System Notes: (1) By convention, the z-axis is the mean location of the north pole (spin axis of Earth) for , (2) the x-axis passes through 0º longitude, (3) the Earth s crust moves slowly with respect to this coordinate system! 6

8 Polar Motion Source: 7

9 Ellipsoid Approximation to Earth s shape Flattened at poles World Geodetic System (WGS)-84 values: Semimajor axis, a = m 1/f = b a f a b a 8

10 Latitude, Longitude, and Height Longitude computation yu arctan, u x 0 u x yu 180arctan, xu 0 and yu 0 xu yu 180arctan, xu 0 and yu 0 xu Latitude and height computation 9

11 Latitude, Longitude, and Height to ECEF acos hcoscos (1 e )tan x ecef asin y ecef hsin cos (1 e )tan z ecef 2 a(1 e )sin hsin e sin 10

12 Geoid Equipotential surface of Earth s gravity field that fits mean sea level on average globally Common representations: Spherical harmonic coefficients Grid of values over Earth Heights measured relative to geoid are referred to as orthometric heights 11

13 Geodetic Height Definitions h = Ellipsoid Height H = Orthometric Height Geoid = Equipotential Gravity Surface N = Geoid Height Ellipsoid = Reference Model Equipotential Surface h(ellipsoid Height) = H(Orthometric Height) + N(Geoid Height) Source: National Geospatial-Intelligence Agency. 12

14 The Geoid Source: National Geospatial-Intelligence Agency. 13

15 Tectonic Plates Source: Jet Propulsion Laboratory. 14

16 Solid Earth Tides and Ocean Loading Earth s surface is not rigid, but rather somewhat pliable Lunar and solar gravity, etc., deform the Earth with strong diurnal and semidiurnal variations over time Displacements may be as large as ~30 cm Ocean tides can cause additional few cm station displacements in coastal locations Station coordinates often defined to exclude effects of solid Earth tides and ocean loading 15

17 Time Difficult to define time! For our purposes, time is the quantity that is read off a clock Any clock includes two main components: Periodic event Counter of that event 16

18 Timescales Universal Time (UT) - one of several time scales based upon rotation rate of Earth UT0 mean solar time based upon astronomical observations from prime meridian UT1 corrects UT0 for polar motion UT2 corrects UT1 for seasonal variations in Earth rotation rate The second used to be defined as 1/86400 of a solar day Atomic time Since 1967, the second has been defined by the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cesium-133 atom International atomic time (TAI) timescale maintained by international collection of atomic clocks 17

19 Sidereal vs Solar Day Sun at ~93M statute miles Star background Day 1 Local Noon Day 2 Local Noon Direction to stellar source Direction to stellar source Day 1 Day 2 Earth rotation Earth rotation Solar Day 24 h Sidereal Day 23 h 56 min 3.5 s 18

20 Coordinated Universal Time (UTC) TAI is a continuous timescale No connection to mean solar time UTC differs from TAI by an integer number of leap seconds Introduced when needed to keep UTC within 0.9 s of UT1 As of May 2013, TAI UTC = 35 s UTC is maintained by the Bureau International des Poids et Mesures (BIPM) in Sevres, France Produced using times maintained by over 250 clocks in 65 nations Not in real-time Official U.S. time is the realization of UTC maintained by the U.S. Naval Observatory UTC(USNO) 19

21 GPS Time GPS Time is maintained by the GPS control segment based upon an average of all system clocks Satellites Monitor stations It is a continuous time scale TAI GPS Time 19 s GPS Time UTC 16 s (as of May 2013) GPS Time is specified with be within 1 microsecond of UTC(USNO) modulo 1 s 20

22 Stability of Frequency Sources - Definitions Consider the oscillator output: Vt () At ()sin(2 ft()) t Instantaneous phase: () t 2 f t() t Instantaneous frequency: f() t Fractional frequency deviation: f d( t) 2 dt 1 d( t) yt () 2 f dt 0 21

23 Frequency Noise Typical characteristics of power spectrum of y(t) are illustrated above. 22

24 Allan Deviation Allan deviation is defined as: where 1 y( ) E( yk 1 yk) 2 2 y k ([ k1] ) ( k) 2 f 0 23

25 Crystal Oscillator Voltage Tuning Varactor Crystal Resonator Amplifier Stabilized output frequency 24

26 Atomic Frequency Standard Quartz Crystal Oscillator VCO Multiplier Feedback o Noise Atomic Resonator Microwave Discriminator Frequency correction signal Output 25

27 Stability of Various Clock Types Source: John Vig IEEE tutorial. 26

28 Kepler s Laws A satellite s orbit is in the shape of an ellipse with the Earth at one focus The radius drawn from the center of the Earth to the satellite sweeps out equal area in equal times The square of the orbital period of a satellite is proportional to the cube of the semi-major axis 27

29 Keplerian Elements Shape of Orbit Keplerian elements: a = semimajor axis e = eccentricity = time of perigee passing A = location of satellite, F = focus (Earth center), P = perigee (lowest point on orbit) 28

30 Keplerian Elements Orientation of Orbit = right ascension of ascending node, i = inclination, = argument of perigee 29

31 Perturbations to Orbits Satellite orbits are not perfect ellipses due to various perturbations: Earth s oblateness Third-body effects sun, moon, etc. Solar radiation pressure As a result, more orbital elements are needed to accurately convey satellite locations Example: Broadcast GPS orbits use basic Keplerian elements plus some rate terms plus sine/cosine corrections to some elements 30

32 Satellite Navigation Positioning For most modern satellite navigation systems, positioning is based upon one-way ranging Ranges to each satellite are estimated using the speed of light multiplied by measured signal transit time (time of receipt minus time of transmission) Most GNSS users do NOT have a high precision clock Thus, ALL range measurements are biased by user clock error multiplied by the speed of light These biased range measurements are commonly referred to as pseudoranges GNSS pseudorange measurements also include many other errors, to be discussed in the next lecture 31

33 Positioning in Two-Dimensions One measurement Three measurements Two measurements 32

34 2D Example Effect of Imperfect User Clock Three measurements 33

35 Positioning in Three Dimensions With three satellites and perfect user clock, one can estimate user location; with imperfect user clock, four satellites are needed to determine user location and clock error (4 equations, 4 unknowns). 34

36 Time and Distance Conversions Speed of light, c = m/s Some conversions: 1 ns ~ 0.3 m 1 s ~ 300 m 1 ms ~ 300 km 1 s ~ 300,000 km 35